L -uniqueness of Schrödinger operators on a Riemannian manifold

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1 L -uniqueness of Schrödinger operators on a Riemannian manifold Ludovic Dan Lemle Abstract. The main purpose of this paper is to study L -uniqueness of Schrödinger operators and generalized Schrödinger operators on a complete Riemannian manifold. Also, we prove the L 1 (E, dµ)-uniqueness of weak solutions for the Fokker-Planck equation associated with this pregenerators..s.c. 2: 47D3, 47F5; 6J6. Key words: C -semigroups; L -uniqueness; Schrödinger operators; Fokker-Planck equation. 1. Preliminaries Let E be a Polish space equipped with a σ-finite measure µ on its Borel σ-field B. It is well known that, for a C -semigroup T (t)} t on L 1 (E, dµ), its adjoint semigroup T (t)} t is no longer strongly continuous on the dual topological space L (E, dµ) of L 1 (E, dµ) with respect to the strong dual topology of L (E, dµ). In [15] Wu and Zhang introduce a new topology on L (E, dµ) for which the usual semigroups in the literature becomes C -semigroups. That is the topology of uniform convergence on compact subsets of (L 1 (E, dµ),. 1 ), denoted by C(L, L 1 ). ore precisely, for an arbitrary point y L (E, dµ), a basis of neighborhoods with respect to C(L, L 1 ) is given by: (1.1) N(y ; K, ε) := y L (E, dµ) sup x K } x, y x, y < ε where K runs over all compact subsets of (L 1 (E, dµ),. 1 ) and ε >. If T (t)} t is a C -semigroup on L 1 (E, µ) with generator L, then T (t)} t is a C -semigroup on ( L (E, dµ), C ( L, L 1)) with generator L. oreover, on can proove that ( L (E, dµ), C ( ( L, L 1)) is complete and that the topological dual of L (E, dµ), C ( L, L 1)) is ( ) L 1 (E, dµ),. 1. Let A : D L (E, dµ) be a linear operator with domain D dense in (L (E, dµ)), with respect to the topology C ( L, L 1). A is said to be a pre-generator in (L (E, dµ)), if there exists some C -semigroup on ( L (E, dµ), C ( L, L 1)) such that its generator L extends A. We say that A is an essential generator in (L (E, dµ)) (or Differential Geometry - Dynamical Systems, Vol. 9, 27, pp c Balkan Society of Geometers, Geometry Balkan Press 27.

2 14 Ludovic Dan Lemle ( L (E, dµ), C ( L, L 1)) -unique), if A is closable and its closure A with respect to C ( L, L 1) is the generator of some C -semigroup on ( L (E, dµ), C ( L, L 1)). This uniqueness notion was studied by Arendt [1], Eberle [5], Djellout [4], Röckner [1], Wu [13] and [14] and others in the Banach spaces setting and Wu and Zhang [15], Lemle and Wu [6] in the case of locally convex spaces. The main result concerning ( L (E, dµ), C ( L, L 1)) -uniqueness of pre-generators is (see [15] or [6] for much more general results) Theorem 1.1. Let A be a linear operator on ( L (E, dµ), C ( L, L 1)) with domain D (the test-function space) which is assumed to be dense in ( L (E, dµ), C ( L, L 1)). Assume that there is a C -semigroup T (t)} t on ( L (E, dµ), C ( L, L 1)) such that its generator L is an extension of A (i.e., A is a pre-generator in ( L (E, dµ), C ( L, L 1)) ). The following assertions are equivalents: (i) A is a ( L (E, dµ), C ( L, L 1)) -essential generator (or ( L (E, dµ), C ( L, L 1)) - unique); (ii) the closure of A in ( L (E, dµ), C ( L, L 1)) is exactly L (i.e., D is a core for L); (iii) A = L which is the generator of the dual C -semigroup T (t)} t on ( L 1 (E, dµ) ) ; (iv) for some λ > ω (ω is the constant in definition of C -semigroup T (t)} t ), the range (λi A)(D) is dense in ( L (E, dµ), C ( L, L 1)) ; (v) (Liouville property) for some λ > ω, Ker (λi A ) = } (i.e., if y D(A ) satisfies (λi A )y =, then y = ); (vi) (uniqueness of solutions for the resolvent equation) for all λ > ω and all y L 1 (E, dµ), the resolvent equation of A (1.2) (λi A )z = y has the unique solution z = ((λi L) 1 ) y = (λi L ) 1 y; (vii) (uniqueness of strong solutions for the Cauchy problem) for each x D ( A ), there is a ( L (E, dµ), C ( L, L 1)) -unique strong solution v(t) = T (t)x of the Cauchy problem (or the Kolmogorov backward equation) t v(t) = Av(t) (1.3) v() = x i.e., t v(t) is differentiable from R + to ( L (E, dµ), C ( L, L 1)) and its derivative t v(t) coincides with Av(t); (viii) (uniqueness of weak solutions for the dual Cauchy problem) for every y L 1 (E, dµ), the dual Cauchy problem (the Kolmogorov forward equation or the Fokker- Planck equation) t u(t) = A u(t) (1.4) u() = y has a ( L 1 (E, dµ) ) -unique weak solution u(t) = T (t)y. ore precisely, there is a( unique function ) R + t u(t) = T (t)y which is continuous from R + to L 1 (E, dµ),. 1 such that (1.5) t x, u(t) y = Ax, u(s) ds, x D;

3 L -uniqueness of Schrödinger operators 15 (ix) there is only one C -semigroup on ( L (E, dµ), C ( L, L 1)) such that its generator extends A. (x) there is only one C -semigroupe on (L 1 (E, dµ), 1 ) such that its generator is contained in A. 2. L -uniqueness of Schrödinger operators on a complete Riemannian manifold Let be a complete Riemannian manifold with volume measure dx. Denote by the Laplace-Beltrami operator with domain C (). The L p (, dx)-uniqueness of C -semigroup generated by (, C ()) is proven by Strichartz [12] for 1 < p <, by Davies [3] for p = 1 and by Li [7] for p =. In fact, Li [7] showed that if is complete and has bounded geometry, then satisfies L 1 -Liouville theorem: if the Ricci curvature of has a negative quadratic lower bound, i.e., if there exists a fixed point and a constant C > such that (2.6) Ric(x) C ( 1 + d(x, ) 2), x where d(x, ) denotes distance from x to o, then every non-negative L 1 -integrable subharmonic function must be constant. Under the same condition, Li also proved that for each h L 1 (, dx), the heat diffusion equation t u(t) = u(t) (2.7) u() = h has one L 1 (, dx)-unique weak solution. Consider the Schrödinger operator (A, C ()) (2.8) A = 2 V where V. In that case, we have Theorem 2.2. Let be a complete Riemmanian manifold such that its Ricci curvature has a negative quadratic lower bound. Then (A, C ()) is L (, dx)- unique with respect to the topology C ( L, L 1). Proof. We prove the theorem by two steps. step 1 (A, C ()) is a pre-generator on ( L (, dx), C ( L, L 1)). Let (B t ) t be the Brownian otion with values in the one-point compactification space defined on some filtered probabilities space (Ω, F, (F t ) t, (B t ) t ) with P x (B = x) = 1 for any initial point x. Let denote by (2.9) τ e = inf t B t = } the explosion time. Consider the Feynman-Kac semigroup (2.1) P V t f(x) = E x 1 [t<τe ]f(b t )e t V (B s )ds.

4 16 Ludovic Dan Lemle Since } Pt V t is a C -semigroup on L 1 (, dx), it is also a C -semigroup on L (, dx) with respect to the topology C ( L, L 1). By Ito s formula, for every f C () it follows that (2.11) f(b t )e t t V (B s)ds f(b ) ( ) 2 V f(b s )e s V (B r)dr ds is a local martingale. As it is bounded over bounded time intervals, it is a true martingale. Thus taking the expectation under P x in the above formula, we find that (2.12) P V t f(x) f(x) = t P V s ( ) 2 V f(x)ds, t which means that f belongs to the domain of the generator L V ( ) of } Pt V t, i.e. L V ( ) extend A. step 2 (A, C ()) is ( L (, dx), C ( L, L 1)) -unique. By Theorem 1.1, it follows that the operator (A, C ()) is L (, dx)-unique if and only if for some λ > ω, the range (λi A) (C ()) is dense in ( L, C ( L, L 1)). It is enough to show that if h L 1 (, dx) satisfies (λi A)h = in the sense of distribution, then h =. Let h L 1 (, dx) such that for some λ > ω (2.13) (λi A)h = in the distribution sense. Then, by Kato s inequality, we have (2.14) h sgn(h) h = 2sgn(h)(λ + V )(h) = 2(λ + V ) h. Thus h is a subharmonic function. By [7, Theorem 1, p. 447] it follows that h is constant, therefore h is constant. Consequently h =. Corollary 2.3. If is a complete Riemannian manifold such that its Ricci curvature has a negative quadratic lower bound, then for every h L 1 (, dx), the Fokker-Planck equation t u(t) = ( 2 V ) u(t) (2.15) u() = h has one L 1 (, dx)-unique weak solution. Proof. By the Theorem 1.1 the L 1 (, dx)-uniqueness of weak solutions for the heat diffusion equation t u(t) = ( 2 V ) u(t) (2.16) u() = h is equivalent to the ( L (, dx), C ( L, L 1)) -uniqueness of the Schrödinger operator (A, C ()) which follows by the Theorem 2.2.

5 L -uniqueness of Schrödinger operators L -uniqueness of generalized Schrödinger operators on a complete Riemannian manifold Consider the generalized Schrödinger operator (or the Nelson s diffusion operator in stochastic mechanics see [14]) (3.17) A = φ with domain C () and an invariant measure dµ φ = e φ(x) dx, where is the Laplace-Beltrami operator and is the gradient operator on and φ C (). The study of this operator is very important because it is well known that A = φ considered as a symmetric operator on L 2 (, e φ(x) dx) is unitary equivalent to the Schrödinger operator H = V, with V = 1 4 φ φ considered on L2 (, dx). This unitary isomorphism U : L 2 (, e φ(x) dx) L 2 (, dx) is given by Uf = e φ 2 f. In the case where = R d or = D is an open domain in R d for the diffusion operator A = + 2 φ φ, L1 (, φ 2 dx)-uniqueness has been studied by Liskevitch [9], Stannat [11] and Wu [14]. The purpose of this section is to study L (, e φ(x) dx)-uniqueness of the operator (A, C ()) using some very recent result of Li [8] concerning the L 1 (, e φ(x) dx) Liouville theorem on a complete Riemannian manifold. Recall that for any constant m n = dim() the symmetric 2-tensor (3.18) Ric m,n (A)(x) = Ric(x) + 2 φ(x) φ(x) φ(x) m n, x is called the Bakry-Emery Ricci curvature of the operator A (see [2]), with the convention that m = n iff A =. The main result of this section is Theorem 3.4. Let be a complete Riemannian manifold, φ C 2 (). Suppose that there exists two constants m > n and C > such that (3.19) Ric m,n (A)(x) C ( 1 + d 2 (x, ) ), x where d(x, ) is the distance from x to a fixed point. Then (A, C ()) is L (, e φ(x) dx)-unique with respect to the topology C ( L, L 1). Proof. We prove the theorem by two steps. step 1 (A, C ()) is a pre-generator on ( L (, e φ(x) dx), C ( L, L 1)). Let (W t ) t be the Riemannian Brownian motion on. The coresponding stochastic differential equation (3.2) dx t = 2dW t (X t )dt admits one unique martigale solution (X t ) t<τe, where τ e is the explosion time. The transition semigroup P t } t is given by the Feynman-Kac formula (3.21) P t f(x) = E x f(x t )1 [t<τe ], f C ()

6 18 Ludovic Dan Lemle Since P t } t is a C -semigroup on (L 1 (, e φ(x) dx), 1 ), it is also a C -semigroup on L (, e φ(x) dx) with respect to the topology C ( L, L 1). By Ito s formula, for every f C () (3.22) f(x t ) f(x) t Af(X t )ds is a local martingale. As it is bounded over bounded time intervals, it is a true martingale. By taking the expectation under P x in the above formula, we find that (3.23) P t f(x) f(x) = t P s Af(x)ds, t which means that f belongs to the domain of the generator L ( ) of P t } t i.e., L ( ) extend A. step 2 (A, C ()) is L (, e φ(x) dx)-unique with respect to the topology C ( L, L 1). By Theorem 1.1, it follows that the operator (A, C ()) is L (, e φ(x) dx)-unique if and only if for some λ > ω, the range (A λi) (C ()) is dense in ( L, C ( L, L 1)). It is enough to show that if u L 1 (, e φ(x) dx) satisfies (A λi)u = in the sense of distribution, then u =. Let u L 1 (, e φ(x) dx) such that (3.24) in the distribution sense, i.e. (3.25) Then from (3.26) (A I)u = u, (A I)f =, f C (). u f e φ(x) dx u φ f e φ(x) dx = uf e φ(x) dx we will obtain the following equality (3.27) u f e φ(x) dx u φ f e φ(x) dx = uf e φ(x) dx for all positive function f H 1,2 (, e φ(x) dx) with compact support. Now on can follow Eberle [5, proof of Theorem 2.5, Step 2] to show that (an inequality of Kato s type) (3.28) u f e φ(x) dx u φ f e φ(x) dx u f e φ(x) dx for all positive function f H 1,2 (, e φ(x) dx) with compact support. But this is equivalent to (3.29) u Af e φ(x) dx u f e φ(x) dx.

7 L -uniqueness of Schrödinger operators 19 That means u is a A-subharmonic function. By the Theorem [8, Theorem 7.1], it follows that u must be a constant function and then u =. By the Theorem 1.1 it follows that (A, C ()) is L (, e φ(x) dx)-unique. Corollary 3.5. In the hypothesis of Theorem 3.4, for every h L 1 (, dx), the Fokker-Planck equation (3.3) t u(t) = A u(t) u() = h has one L 1 (, dx)-unique weak solution. Proof. The assertion follows by Theorem 1.1 and the Theorem 3.4. Acknowledgements. The author is very grateful to Professor Liming WU for his kind invitation to Wuhan University, China, during ay-june 26. This work is partially supported by Yangtze Research Programme, Wuhan University, China, and the Town Council of Hunedoara, Romania. References [1] W. Arendt The abstract Cauchy problem, special semigroups and perturbation. In R. Nagel (Ed.), One Parameter Semigroups of Positive Operators. Lect. Notes in ath., vol. 1184, Springer, Berlin, [2] D. Bakry,. Emery Diffusion hypercontractives. Lect. Notes in ath., 1123 (1985), [3] E.B. Davies L 1 -properties of second order elliptic operators. Bull. London ath. Soc., 17 (1985), [4] H. Djellout Unicité dans L p d opérateurs de Nelson. Preprint, [5] A. Eberle Uniquenees and non-uniqueness of singular diffusion operators. Doctorthesis, Bielefeld, [6] D.L. Lemle, L. Wu Uniqueness of a pre-generator for C -semigroup on a general locally convex vector space. Preprint, 26. [7] P. Li Uniqueness of L 1 solution for the Laplace equation and the heat equation on Riemannian manifolds. J. Diff. Geom., 2 (1984), [8] X.D. Li Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. ath. Pures Appl., 84 (25), [9] V. Liskevitch On the uniqueness problem for Dirichlet operators. J. Funct. Anal., 162 (1999), [1]. Röckner L p -analysis of finite and infinite dimensional diffusion operators. Lect. Notes in ath., 1715 (1998), [11] W. Stannat (Nonsymmetric) Dirichlet operators on L 1 : existence, uniqueness and associated arkov processes. Ann. Scuola Norm. Sup. Pisa, 28 (1999), [12] R.S. Strichartz Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal., 52 (1983), [13] L. Wu Uniqueness of Schrödinger Operators Restricted in a Domain. J. Funct. Anal., 153 (1998),

8 11 Ludovic Dan Lemle [14] L. Wu Uniqueness of Nelson s diffusions. Probab. Theory Relat. Fields, 114 (1999), [15] L. Wu, Y. Zhang A new topological approach to the L -uniqueness of operators and L 1 -uniqueness of Fokker-Planck equations. J. Funct. Anal., 241 (26), Author s address: Ludovic Dan Lemle Laboratoire de athématiques, CNRS-UR 662, Université Blaise Pascal, Aubière, France lemle.dan@fih.upt.ro